AST-03

Bondi accretion

Ṁ = 4π λ ρ (G M)² / c_s³. Spherical accretion onto a point mass.

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GravityBondi

Governing equation

\dot M=4\pi\lambda\rho (GM)^2/c_s^3

$M \in \left[0.2, 100\right]$

where

M: Mass (M_\odot)
\rho: Density (10⁻²⁰ kg/m³)
c_s: Sound speed (km/s)
\lambda: Bondi λ (—)
\dot M: Accretion rate (M_\odot/yr)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\dot M=4\pi\lambda\rho (GM)^2/c_s^3

$M \in \left[0.2, 100\right]$

$M = 0.2$ $\dot M = 1.41\times 10^{-15}$

Along the curve

②

Domain of the variable — blue

M_{\min} = 0.2,\quad M_{\max} = 100

$M \in \left[0.2, 100\right]$

$M = 0.2$ $M = 0.2$

Along the curve

③

Response curve — green

\dot M=4\pi\lambda\rho (GM)^2/c_s^3

$M \in \left[0.2, 100\right]$

$M = 0.2$ $\dot M = 1.41\times 10^{-15}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\dot M}{\mathrm{d}M}

$M \in \left[0.2, 100\right]$

$M = 0.828$ $\dot M^{\prime} = 5.81\times 10^{-14}$

Along the curve

Lecture brief

Historical brief

Hubble expansion, Jeans collapse, Eddington luminosity, Bondi accretion and Stefan–Boltzmann stars are the first astrophysical budgets. The sheets scale a star, a cloud and a horizon. This sheet (AST-03 — Bondi accretion) is the form associated with Bondi. Working symbols: $M$ , $\rho$ , $c_s$ , $\lambda$ $\rightarrow$ $\dot M$ . λ ~ 1 for γ = 5/3. The Bondi radius is G M / c_s².

Purpose

Purpose: compute

\dot M

from

M

\rho

c_s

\lambda

in Astrophysics via

\dot M=4\pi\lambda\rho (GM)^2/c_s^3

Ṁ = 4π λ ρ (G M)² / c_s³. Spherical accretion onto a point mass. Use it when a real astrophysics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

M = 10.000\,\mathrm{M_odot}

\rho = 1.000\,\mathrm{10⁻^{2}⁰ kg/m^{3}}

c_s = 10.000\,\mathrm{km/s}

\lambda = 1.000\,\mathrm{—}

, the governing relation

\dot M=4\pi\lambda\rho (GM)^2/c_s^3

yields

\dot M = 3.513e-12\,\mathrm{M_odot/yr}

. A star, an inward spherical flow. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Mass M10 M_\odotDensity \rho1 10⁻²⁰ kg/m³Sound speed c_s10 km/sBondi λ \lambda1 —

Outputs

Accretion rate \dot M0.0000 M_\odot/yr

Reading speed

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Narration of this film

A star, an inward spherical flow.

λ ~ 1 for γ = 5/3. The Bondi radius is G M / c_s².

Reading speed

Watch on YouTube